

Introduction to Logic

Tools for Thought


The treatment of semantics in Logic is similar to its treatment in Algebra. Algebra is unconcerned with the realworld significance of variables. What is interesting are the relationships among the values of the variables expressed in the equations we write. Algebraic methods are designed to respect these relationships, independent of what the variables represent.
In a similar way, Logic is unconcerned with the real world significance of proposition constants. What is interesting is the relationship among the truth values of simple sentences and the truth values of compound sentences within which the simple sentences are contained. As with Algebra, logical reasoning methods are independent of the significance of proposition constants; all that matter is the form of sentences.

Although the values assigned to proposition constants are not crucial in the sense just described, in talking about Logic, it is sometimes useful to make truth assignments explicit and to consider various assignments or all assignments and so forth. Such an assignment is called a truth assignment.
Formally, a truth assignment for a propositional vocabulary is a function assigning a truth value to each of the proposition constants of the vocabulary. In what follows, we use the digit 1 as a synonym for true and 0 as a synonym for false; and we refer to the value of a constant or expression under a truth assignment i by superscripting the constant or expression with i as the superscript.

The assignment shown below is an example for the case of a propositional vocabulary with just three proposition constants, viz. p, q, and r.
p^{i} = 1
q^{i} = 0
r^{i} = 1
The following assignment is another truth assignment for the same vocabulary.
p^{i} = 0
q^{i} = 0
r^{i} = 1

Note that the formulas above are not themselves sentences in Propositional Logic. Propositional Logic does not allow superscripts and does not use the = symbol. Rather, these are informal, metalevel statements about particular truth assignments. Although talking about Propositional Logic using a notation similar to that Propositional Logic can sometimes be confusing, it allows us to convey metainformation precisely and efficiently. To minimize problems, in this book we use such metanotation infrequently and only when there is little chance of confusion.

Looking at the preceding truth assignments, it is important to bear in mind that, as far as logic is concerned, any truth assignment is as good as any other. Logic itself does not fix the truth assignment of individual proposition constants.
On the other hand, given a truth assignment for the proposition constants of a language, logic does fix the truth assignment for all compound sentences in that language. In fact, it is possible to determine the truth value of a compound sentence by repeatedly applying the following rules.

If the truth value of a sentence is true, the truth value of its negation is false. If the truth value of a sentence is false, the truth value of its negation is true.

The truth value of a conjunction is true if and only if the truth values of its conjuncts are both true; otherwise, the truth value is false.
φ 
ψ 
φ ∧ ψ 
1 
1 
1 
1 
0 
0 
0 
1 
0 
0 
0 
0 

The truth value of a disjunction is true if and only if the truth value of at least one its disjuncts is true; otherwise, the truth value is false. Note that this is the inclusive or interpretation of the ∨ operator and is differentiated from the exclusive or interpretation in which a disjunction is true if and only if an odd number of its disjuncts are true.
φ 
ψ 
φ ∨ ψ 
1 
1 
1 
1 
0 
1 
0 
1 
1 
0 
0 
0 

The truth value of an implication is false if and only if its antecedent is true and its consequent is false; otherwise, the truth value is true. This is called material implication.
φ 
ψ 
φ ⇒ ψ 
1 
1 
1 
1 
0 
0 
0 
1 
1 
0 
0 
1 

A biconditional is true if and only if the truth values of its constituents agree, i.e. they are either both true or both false.
φ 
ψ 
φ ⇔ ψ 
1 
1 
1 
1 
0 
0 
0 
1 
0 
0 
0 
1 

Using these definitions, it is easy to determine the truth values of compound sentences with proposition constants as constituents. As we shall see in the next section, we can determine the truth values of compound sentences with other compound sentences as parts by first evaluating the constituent sentences and then applying these definitions to the results.
We finish up this section with a few definitions for future use. We say that a truth assignment satisfies a sentence if and only if the sentence is true under that truth assignment. We say that a truth assignment falsifies a sentence if and only if the sentence is false under that truth assignment. A truth assignment satisfies a set of sentences if and only if it satisfies every sentence in the set. A truth assignment falsifies a set of sentences if and only if it falsifies at least one sentence in the set.

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